Codimension-one and codimension-two bifurcations in a new discrete chaotic map based on gene regulatory network model

In this paper, the stability and bifurcations of a new discrete chaotic map based on gene regulatory network are studied. Firstly, the existence and stability conditions of the fixed points are given. Secondly, the conditions for existence of three cases of codimension-one bifurcations (fold bifurcation, flip bifurcation and Neimark–Sacker bifurcation) are derived by using the center manifold theorem and bifurcation theory. Then, the conditions for the occurrence of codimension-two bifurcation (fold–flip bifurcation, 1:2, 1:3 and 1:4 strong resonance) are investigated by using several variable substitutions and introduction of new parameters. Meanwhile, these bifurcation curves are returned to the original variables and parameters to express for easy verification. The corresponding numerical simulations and numerical continuation results not only show the validity of the proposed results, but also exhibit the interesting and complex dynamical behaviors. Finally, some initial conditions and two-parameter spaces analysis are given numerically. The local attraction basins and two-parameter space plots display interesting dynamical behaviors of the discrete system operating with different integral step size and other parameters changing.